Small Strain Elasticity Research Articles

Development of static and kinematic approaches in geomechanical problems

The hypothesis of a linear stress field of a rock mass undisturbed by driving workings underlies the solution to many problems of rock mechanics. The problem of rock pressure control requires knowledge of the stress-strain state of the rock mass in the vicinity of workings, which causes the development of analytical and numerical methods for calculating the deformation. When the methods of numerical calculation were not sufficiently developed, the analytical capabilities of the selection of stress functions were used, which satisfied the conditions of small strain elasticity theory. The greatest success in this direction was achieved for the areas with angular points. The development of numerical methods shifted the point of influence of investigations to calculating the deformation of specific cases arising in a rock mass with workings. It is established in the work that analytical solutions for the areas with angular points are incorrect, and numerical solutions of Cauchy problems are also incorrect if an additional problem is not considered.

Global/local analytical and numerical free vibration analysis of sandwich columns

Sandwich structures are increasingly applied in many industrial fields of application, due to their lightweight combined with favorable mechanical properties. Despite this, such structures are subject to specific failure modes, such as buckling or vibratory resonance. In both cases, due to the presence of thin and stiff skins, global but also local modes may be of great interest when dimensioning such composite structures, which makes it impossible to use classical models. In this paper, the free vibration problem of classical sandwich columns (with homogeneous core materials) is investigated, using special kinematic models, so as to deal with both global and local eigenmodes in an effective and precise way. First, the problem is addressed analytically, where the two faces are represented by Euler-Bernoulli beams and the core material is considered as a 2D continuous solid, in small strain elasticity. Then, an enriched 1D finite element formulation is developed, so as to handle efficiently more general configurations encountered in practice. The homogeneous core layer is here described using hyperbolic functions, in accordance with the modal displacement fields obtained analytically. The present analytical and numerical solutions (natural frequencies and vibration modes) are contrasted against each other and compared to 2D reference numerical results.

A Hybrid High-Order Discretization Combined with Nitsche's Method for Contact and Tresca Friction in Small Strain Elasticity

We devise and analyze a Hybrid high-order (HHO) method to discretize unilateral and bilateral contact problems with Tresca friction in small strain elasticity. The nonlinear frictional contact conditions are enforced weakly by means of a consistent Nitsche's technique with symmetric, incomplete, and skew-symmetric variants. The present HHO-Nitsche method supports polyhedral meshes and delivers optimal energy-error estimates for smooth solutions under some minimal thresholds on the penalty parameters for all the symmetry variants. An explicit tracking of the dependency of the penalty parameters on the material coefficients is carried out to identify the robustness of the method in the incompressible limit, showing the more advantageous properties of the skew-symmetric variant. 2D and 3D numerical results including comparisons to benchmarks from the literature and to solutions obtained with an industrial software, as well as a prototype for an industrial application, illustrate the theoretical results and reveal that in practice the method behaves in a robust manner for all the symmetry variants in Nitsche's formulation.

Existence and asymptotic results for an intrinsic model of small-strain incompatible elasticity

A general model of incompatible small-strain elasticity is presented and analyzed, based on the linearized strain and its associated incompatibility tensor field. Strain incompatibility accounts for the presence of dislocations, whose motion is ultimately responsible for the plastic behaviour of solids. The specific functional setting is built up, on which existence results are proved. Our solution strategy is essentially based on the projection of the governing equations on appropriate subspaces in the spirit of the Leray decomposition of solenoidal square-integrable velocity fields in hydrodynamics. It is also strongly related with the Beltrami decomposition of symmetric tensor fields in the wake of previous works by the authors. Moreover a novel model parameter is introduced, the incompatibility modulus, that measures the resistance of the elastic material to incompatible deformations. An important result of our study is that classical linearized elasticity is recovered as the limit case when the incompatibility modulus goes to infinity. Several examples are provided to illustrate this property and the physical meaning of the incompatibility modulus in connection with the dissipative nature of the processes under consideration.

Skew-symmetric Nitsche’s formulation in isogeometric analysis: Dirichlet and symmetry conditions, patch coupling and frictionless contact

A simple skew-symmetric Nitsche’s formulation is introduced into the framework of isogeometric analysis (IGA) to deal with various problems in small strain elasticity: essential boundary conditions, symmetry conditions for Kirchhoff plates, patch coupling in statics and in modal analysis as well as Signorini contact conditions. For linear boundary or interface conditions, the skew-symmetric formulation is parameter-free. For contact conditions, it remains stable and accurate for a wide range of the stabilization parameter. Several numerical tests are performed to illustrate its accuracy, stability and convergence performance. We investigate particularly the effects introduced by Nitsche’s coupling, including the convergence performance and condition numbers in statics as well as the extra “outlier” frequencies and corresponding eigenmodes in structural dynamics. We present the Hertz test, the block test, and a 3D self-contact example showing that the skew-symmetric Nitsche’s formulation is a suitable approach to simulate contact problems in IGA.

A study on the vibrational properties of weight-efficient plates made of material with functionally graded porosity

Functionally graded (FG) porous structures are a novel class of weight-efficient engineering materials and characterized by introducing graded non-uniform porosities. Due to the wide range of applications of porous materials in lightweight structures, biomedical systems and separation processes, in this paper, three-dimensional (3D) free vibration analyses of light-weight plates made of uniform and non-uniform graded porous materials resting on two-parameter elastic foundations are investigated. The mathematical formulation is based on linear and small strain elasticity assumptions. The plate with FG porosity is simply supported at the edges and is assumed to have three variations of porosities through the thickness. A semi-analytical approach composed of differential quadrature method (DQM) and series solution is adopted to solve the equations of motion. Validation and convergence studies are also done to demonstrate the accuracy of the results. The natural frequencies of the simply-supported plate made of material with functionally graded porosity are calculated and presented in both tabular and graphical forms. The effects of different porosity distributions, porosity parameter, elastic coefficients of foundation and geometrical parameters on the results are investigated. The results of the present analysis indicate the advantages of using a plate with graded porosity over a traditional plate with uniform porosity.

Data-based derivation of material response

This paper proposes a method to identify the strain–stress relation of non-linear elastic materials, without any underlying constitutive equation. The approach is based on the concept of Data Driven Computational Mechanics recently introduced by Kirchdoerfer and Ortiz (2016). From a collection of non-homogeneous strain fields, for example measured through Digital Image Correlation, the algorithm builds a database of strain–stress couples that sample the mechanical response of the material for the range of measured strains. The method is first derived for truss structures and then extended to the case of small-strain elasticity. The method accuracy, sensitivity to measurement noise and parameters are discussed using manufactured data.

Revisiting the Constrained Blister Test to Measure Thin Film Adhesion

A thin film is clamped at the periphery to form a circular freestanding diaphragm before a uniform pressure, p, is applied to inflate it into a blister. The bulging membrane adheres to a rigid constraining plate with height, w0, from the nondeformed membrane. Increasing pressure expands the contact circle of radius, c. Depressurization causes shrinkage of the contact and “pull-off” or spontaneous detachment from the plate. Simultaneous measurement of (p, w0, c) allows one to determine the adhesion energy, γ. A solid mechanics model is constructed based on small strain and linear elasticity, which shows a characteristic loading–unloading hysteresis. The results are consistent with a large deformation model in the literature.

A discontinuous Galerkin method for cohesive zone modelling

We propose a discontinuous finite element method for small strain elasticity allowing for cohesive zone modeling. The method yields a seamless transition between the discontinuous Galerkin method and classical cohesive zone modeling. Some relevant numerical examples are presented.

Two-Dimensional Differential Quadrature Solution for Vibration Characteristics of Two-Dimensional Functionally Graded Metal/Ceramic Open Cylindrical Shells

In this article, semi-analytical 3D elasticity solutions are presented to study the vibration characteristics of two-dimensional functionally graded (2D FGM) metal/ceramic open cylindrical shells under various boundary conditions. In the present formulation, the shell has a smooth variation of volume fractions of metal and ceramic in the radial and axial directions with power law functions. The study is carried out based on the three-dimensional, linear and small strain elasticity theory. The open cylindrical shell is assumed to be simply supported at one pair of opposite edges and arbitrary boundary conditions at the other edges such that trigonometric functions expansion can be used to satisfy the boundary conditions precisely at simply supported edges. The two-dimensional generalized differential quadrature method (GDQM) as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated and to validate the results, comparisons are made with the available solutions for FGM cylindrical shells. Results indicate the advantages of using functionally graded open cylindrical shells with graded volume fractions in two directions to a more flexible design than the conventional one-dimensional FGM. The effects of different boundary conditions, various geometrical parameters and volume fraction indices in radial and axial directions on the vibration characteristics of 2D FGM open cylindrical shells are investigated. Finally, frequency parameters of 2D FGM open cylindrical shells are compared with radially and axially conventional FGM.

Finite element analysis of an inflatable torus considering air mass structural element

Inflatable structures, also known as gossamer structures, are at high boom in the current space technology due to their low mass and compact size comparing to the traditional spacecraft designing. Internal pressure becomes the major source of strength and rigidity, essentially stiffen the structure. However, inflatable space based membrane structure are at high risk to the vibration disturbance due to their low structural stiffness and material damping. Hence, the vibration modes of the structure should be known to a high degree of accuracy in order to provide better control authority. In the past, most of the studies conducted on the vibration analysis of gossamer structures used inaccurate or approximate theories in modeling the internal pressure. The toroidal shaped structure is one of the important key element in space application, helps to support the reflector in space application. This paper discusses the finite-element analysis of an inflated torus. The eigen-frequencies are obtained via three-dimensional small-strain elasticity theory, based on extremum energy principle. The two finite-element model (model-1 and model-2) have cases have been generated using a commercial finite-element package. The structure model-1 with shell element and model-2 with the combination of the mass of enclosed fluid (air) added to the shell elements have been taken for the study. The model-1 is computed with present analytical approach to understand the convergence rate and the accuracy. The convergence study is made available for the symmetric modes and anti-symmetric modes about the centroidal-axis plane, meeting the eigen-frequencies of an inflatable torus with the circular cross section. The structural model-2 is introduced with air mass element and analyzed its eigen-frequency with different aspect ratio and mode shape response using in-plane and out-plane loading condition are studied.

Three dimensional mechanical buckling of FG plates with general boundary conditions

The mechanical buckling analysis of rectangular functionally graded plates under different axial loadings is considered. The analysis is based on the small strain elasticity theory with different boundary conditions. The material properties of the plate vary through the thickness direction according to a simple power law. Three dimensional buckling solutions are obtained using the Ritz method with Chebyshev polynomials as assumed displacement functions. The convergence and comparison studies are presented and effects of the different material composition and the plate geometry (side–side ratio, side–thickness ratio) on the critical buckling loads and mode shapes are investigated.

Effect of continuously grading fiber orientation face sheets on vibration of sandwich panels with FGM core

The main purpose of this paper is to investigate effect of continuously grading fiber orientation face sheets on free vibration of sandwich panels with functionally graded core using generalized power-law distribution. The benefit of using generalized power-law distribution is to illustrate and present useful results arising from symmetric, asymmetric and classic profiles. The fiber-reinforced composite material studied in the present work consists of silicon carbide fibers embedded in titanium matrix with the fiber volume fraction and fiber orientation graded in the radial direction. The face sheets have variation of the fiber orientation while the core has variation of fiber volume fraction. Generalized differential quadrature (GDQ) method is used to yield natural frequencies of the simply supported functionally graded sandwich (FGSW) panels on the basis of the 2-D, linear and small strain elasticity theory. The fast rate of convergence of the method is demonstrated and comparison studies are carried out to establish its very high accuracy and versatility. In this research work, a detailed parametric study is carried out to highlight the influences of continuously grading fiber orientation face sheets and different profile of fiber volume fraction and fiber orientation on the vibration characteristics of the FGSW panels.

A novel 2-D six-parameter power-law distribution for free vibration and vibrational displacements of two-dimensional functionally graded fiber-reinforced curved panels

As a first endeavor, the three-dimensional free vibration and vibrational displacements characteristics of two-dimensional functionally graded fiber-reinforced (2-D FGFR) curved panels with different boundary conditions are presented. This paper presents a novel 2-D six-parameter power-law distribution for fiber volume fractions of 2-D FGFR that gives designers a powerful tool for design flexible of structures under multi-functional requirements. Various material profiles in two radial and axial directions can be illustrated using the six-parameter power-law distribution. The study is carried out based on the three-dimensional, linear and small strain elasticity theory . In this work, orthotropic panel is assumed to be simply supported at one pair of opposite edges and arbitrary boundary conditions at the other edges such that trigonometric functions expansion can be used to satisfy the boundary conditions precisely at simply supported edges . The 2-D generalized differential quadrature method (GDQM) as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated and to validate the results, comparisons are made with the available solutions for FGM curved panels. Results indicate by using the 2-D six-parameter power-law distribution, it is possible to study the influence of different kinds of two-directional material profiles including symmetric and classic on the natural frequencies and modal displacements of a 2-D FGFR panel. Furthermore, maximum amplitude and uniformity of modal displacements distributions can be modified to a required manner by selecting suitable different parameters of 2-D power-law distribution and several various volume fractions profiles in two directions. ► We present a 3-D elasticity solution for the analysis of the natural frequencies and modal displacements components of 2-D FGFR curved panels. ► The novel power-law distribution proposed gives designers a powerful tool for design flexible of structures under multi-functional requirements. ► The obtained results indicate the advantages of using graded fiber volume fractions in two directions to a more flexible design than the conventional FGFR.

Three-dimensional free vibration of variable thickness thick circular and annular isotropic and functionally graded plates on Pasternak foundation

This paper describes a study of three-dimensional free vibration analysis of thick circular and annular isotropic and functionally graded (FG) plates with variable thickness along the radial direction, resting on Pasternak foundation. The formulation is based on the linear, small strain and exact elasticity theory. Plates with different boundary conditions are considered and the material properties of the FG plate are assumed to vary continuously through the thickness according to power law. The kinematic and the potential energy of the plate–foundation system are formulated and the polynomial-Ritz method is used to solve the eigenvalue problem. Convergence and comparison studies are done to demonstrate the correctness and accuracy of the present method. With respect to geometric parameters, elastic coefficients of foundation and different boundary conditions some new results are reported which may be used as benchmark solutions for future researches.

Free vibration analysis of continuous grading fiber reinforced plates on elastic foundation

The free vibration characteristics of rectangular continuous grading fiber reinforced (CGFR) plates resting on elastic foundations have been studied, based on the three-dimensional, linear and small strain elasticity theory. The foundation is described by the Pasternak or two-parameter model. The CGFR plate is simply supported at the edges and is assumed to have an arbitrary variation of fiber volume fraction in the thickness direction. Suitable displacement functions that identically satisfy the simply supported boundary conditions are used to reduce the equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which can be solved by differential quadrature method (DQM) to obtain natural frequencies. Convergence studies have been performed on CGFR plates on the elastic foundations. It is shown that the present method has a rapid convergent rate, stable numerical operation and very high accuracy. Besides results for CGFR plate with arbitrary variation of fiber volume fraction in the plate’s thickness are compared with discrete laminated composite plate. The main contribution of this work is to present useful results for continuous grading of fiber reinforcement in the thickness direction of a plate on elastic foundation and comparison with similar discrete laminate composite plate. Results indicate the advantages of using CGFR plate with graded fiber volume fractions over traditional discretely laminated plates.

Maximum-entropy meshfree method for compressible and near-incompressible elasticity

Numerical integration errors and volumetric locking in the near-incompressible limit are two outstanding issues in Galerkin-based meshfree computations. In this paper, we present a modified Gaussian integration scheme on background cells for meshfree methods that alleviates errors in numerical integration and ensures patch test satisfaction to machine precision. Secondly, a locking-free small-strain elasticity formulation for meshfree methods is proposed, which draws on developments in assumed strain methods and nodal integration techniques. In this study, maximum-entropy basis functions are used; however, the generality of our approach permits the use of any meshfree approximation. Various benchmark problems in two-dimensional compressible and near-incompressible small strain elasticity are presented to demonstrate the accuracy and optimal convergence in the energy norm of the maximum-entropy meshfree formulation.

A mortar-finite element approach to lubricated contact problems

In this paper, a new numerical approach, based on mortar discretization of interface fields, is proposed to solve lubricated contact problems between deformable solid bodies. The Reynolds approximation to the thin film fluid equations is used to describe the fluid behavior on the interface, while the equations of small strain elasticity (to which finite elements are applied) are assumed to govern the solid body behavior. The result of this approach is a monolithic scheme in which the fluid and solid mechanics are solved simultaneously, without the need for iterative or staggered approaches between the fluid and solid phases. Key features of the approach are that the fluid film thickness, directly related to the deformation of the solid phase, is computed from a least squares projection based on dual basis functions (as pioneered by Wohlmuth [B.I. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition, Springer-Verlag, Heidelberg, 2001]). The free boundary problem for the fluid phase, corresponding to the boundary between lubricated and cavitated regions, is regularized with a penalty method and thus resolved as part of the solution. Several numerical examples are given to demonstrate the approach.

Three-dimensional vibration analysis of prisms with isosceles triangular cross-section

This paper studies the three-dimensional (3-D) free vibration of uniform prisms with isosceles triangular cross-section, based on the exact, linear and small strain elasticity theory. The actual triangular prismatic domain is first mapped onto a basic cubic domain. Then the Ritz method is applied to derive the eigenfrequency equation from the energy functional of the prism. A set of triplicate Chebyshev polynomial series, multiplied by a boundary function chosen to, a priori, satisfy the geometric boundary conditions of the prism is developed as the admissible functions of each displacement component. The convergence and comparison study demonstrates the high accuracy and numerical robustness of the present method. The effect of length-thickness ratio and apex angle on eigenfrequencies of the prisms is studied in detail and the results are compared with those obtained from the classical one-dimensional theory and the 3-D finite element method. Sets of valuable data known for the first time are reported, which can serve as benchmark values in applying various approximate beam and rod theories.

Three-dimensional vibration analysis of cantilevered skew plates

Three-dimensional (3-D) free vibration of cantilevered thick skew plates is analyzed, based on the exact, linear and small-strain elasticity theory. The skew domain of the plate is mapped onto a cubic domain. A set of triplicate Chebyshev polynomials multiplied by a boundary function is developed as the trial functions of each displacement components. Using the Ritz method, the eigenvalue equation is derived from the strain energy and kinetic energy of the plate. The vibration modes are divided into the antisymmetric and symmetric ones in the thickness direction, therefore, can be studied individually. The convergence study shows that the first eight frequency parameters for each mode categories can be obtained with an accuracy of at least four significant figures. The effect of geometric parameters, such as skew angle, aspect ratio and span-thickness ratio, on frequency parameters is studied. The results are compared with those obtained by using the algebraic polynomials as trial functions and the 3-D finite element solutions, respectively. It is shown that the Chebyshev polynomials can provide better numerical stability than the algebraic polynomials, especially for the plates with large skew angle. The present results can serve as the benchmark data for the accuracy evaluation of other computational techniques.